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Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

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Abstract

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero.

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References

  1. Chern I.-L.: Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws. Commun. Math. Phys. 138, 51–61 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Cole J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math. 9, 225–236 (1951)

    MATH  MathSciNet  Google Scholar 

  3. Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal. 95, 325–344 (1986)

    Article  MATH  Google Scholar 

  4. Hopf E.: The partial differential equation u t  + uu x  = μ u xx . Comm. Pure Appl. Math. 3, 201–230 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lax P.D.: Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10, 537–566 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(328), v+108 pp. (1985)

  7. Liu, T.-P.: Interactions of nonlinear hyperbolic waves. In: Nonlinear Analysis, Liu, F.-C., Liu, T.-P. (eds)., Singapore: World Scientific, 1991, pp. 171–184

  8. Liu T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50, 1113–1182 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 125(599), viii+120 pp. (1997)

  10. Liu T.-P., Zeng Y.: Compressible Navier-Stokes equations with zero heat conductivity. J. Diff. Eqs. 153, 225–291 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, T.-P., Zeng, Y.: Fundamental solution for hyperbolic-parabolic system around a shock profile. Preprint

  12. Matsumura A., Nishihara K.: On a stability of travelling wave solution of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 3, 1–13 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Shu C.-W., Zeng Y.: High-order essentially non-oscillatory scheme for viscoelasticity with fading memory. Quart. Appl. Math. 55, 459–484 (1997)

    MATH  MathSciNet  Google Scholar 

  14. Smoller J.: Shock Waves and Reaction-Diffusion Equations 2nd ed. Springer-Verlag, New York (1994)

    MATH  Google Scholar 

  15. Szepessy A., Xin Z.P.: Nonlinear stability of viscous shock waves. Arch. Rat. Mech. Anal. 122, 53–103 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zeng Y.: L 1 asymptotic behavior of compressible, isentropic, viscous 1-D flow. Comm. Pure Appl. Math. 47, 1053–1082 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zeng Y.: L p asymptotic behavior of solutions to hyperbolic-parabolic systems of conservation laws. Arch. Math. (Basel) 66, 310–319 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Zeng Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rat. Mech. Anal. 150, 225–279 (1999)

    Article  MATH  Google Scholar 

  19. Zeng, Y.: Gas Flows with Several Thermal Nonequilibrium Modes. Arch. Rat. Mech. Anal. (to appear)

  20. Zumbrum, K.: Planar stability criteria for viscous shock waves of systems with real viscosity. In: Hyperbolic Systems of Balance Laws, Lecture Notes in Math. 1911, Berlin: Springer, 2007, pp. 229–326

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, Taiwan

    Tai-Ping Liu

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Tai-Ping Liu

  3. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    Yanni Zeng

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  1. Tai-Ping Liu
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  2. Yanni Zeng
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Correspondence to Yanni Zeng.

Additional information

Communicated by P. Constantin

The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248.

The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.

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Liu, TP., Zeng, Y. Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile. Commun. Math. Phys. 290, 23–82 (2009). https://doi.org/10.1007/s00220-009-0820-6

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  • DOI: https://doi.org/10.1007/s00220-009-0820-6

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